scholarly journals A matroid associated with a phylogenetic tree

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Andreas Dress ◽  
Katharina Huber ◽  
Mike Steel
Keyword(s):  
Phylogenetic Tree ◽  
Edge Weight ◽  
Canonical Isomorphism ◽  
Special Issue ◽  
Finite Tree ◽  
Tree Metric ◽  
Negative Edge ◽  
Path Distance ◽  
Finite Set ◽  
International Audience

Special issue PRIMA 2013 International audience A (pseudo-)metric D on a finite set X is said to be a \textquotelefttree metric\textquoteright if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is – up to canonical isomorphism – uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree\textquoterights edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (\textquoteleftlasso\textquoteright) these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the \textquotelefttight edge-weight lassos\textquoteright for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Special Issue on Visualisations in Historical Linguistics: Introduction

2020 ◽  
Vol Special issue on... ◽  
Author(s):  
Benjamin Molineaux ◽  
Bettelou Los ◽  
Martti Mäkinen
Keyword(s):  
Historical Linguistics ◽  
Data Visualisation ◽  
Complex Materials ◽  
Special Issue ◽  
The Core ◽  
Methodological Research ◽  
International Audience ◽  
Introductory Paper ◽  
The Individual ◽  
Historical Periods

International audience The advent of ever-larger and more diverse historical corpora for different historical periods and linguistic varieties has led to the impossibility of obtaining simple, direct-and yet balancedrepresentations of the core patterns in the data. In order to draw insights from heterogeneous and complex materials of this type, historical linguists have begun to reach for a growing number of data visualisation techniques, from the statistical, to the cartographical, the network-based and beyond. An exploration of the state of this art was the objective of a workshop at the 2018 International Conference on English Historical Linguistics, from whence most of the materials of this Special Issue are drawn. This brief introductory paper outlines the background and relevance of this line of methodological research and presents a summary of the individual papers that make up the collection.

scholarly journals On the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals The extended equivalence and equation solvability problems for groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Gabor Horvath ◽  
Csaba Szabo
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Nilpotent Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Solvability Problem ◽  
Np Complete ◽  
Equation Solvability

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.

scholarly journals Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Thomas P. Hayes
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Communication Complexity ◽  
60Th Birthday ◽  
Equal Size ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Zarankiewicz Problem ◽  
Complexity Hierarchy

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

scholarly journals Grammatical compression: compressed equivalence and other problems

2010 ◽  
Vol Vol. 12 no. 4 ◽  
Author(s):  
Alberto Bertoni ◽  
Roberto Radicioni
Keyword(s):  
Strategic Games ◽  
Special Issue ◽  
Ability To Work ◽  
Inclusion Problems ◽  
Polynomial Time Algorithms ◽  
Straight Line ◽  
International Audience ◽  
Context Free ◽  
Compressed Data ◽  
Straight Line Programs

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience In this work, we focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs on a given algebra. In our exposition, we try to survey general results by presenting some meaningful examples; moreover, where possible, we outline the proofs in order to give an insight of the methods and the techniques. We recall some recent results for the problem PosSLP, consisting of deciding if the integer defined by a straight-line program on the ring Z is greater than zero; we discuss some implications in the areas of numerical analysis and strategic games. Furthermore, we propose some methods for reducing Compressed Word Problem from an algebra to another; reductions from trace monoids to the semiring of nonnegative integers are exhibited and polynomial time algorithms for compressed equivalence in monoids related to Dyck reductions are shown. Finally, we consider inclusion problems for context-free languages, proving how in some cases efficient algorithms for these problems benefit from the ability to work with compressed data.

scholarly journals Tree Reconstruction from Triplet Cover Distances

10.37236/3388 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Katharina T. Huber ◽  
Mike Steel
Keyword(s):  
Polynomial Time ◽  
Classical Result ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Interior Vertex ◽  
Evolutionary Genomics ◽  
Tree Reconstruction ◽  
Finite Tree ◽  
Path Distance ◽  
Special Case

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset $\mathcal{L}$ of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in $\mathcal{L}$. It is known that any set ${\mathcal L}$ with this property for a tree in which all interior vertices have degree 3 must form a cover  for $T$ - that is, for each interior vertex $v$ of $T$, ${\mathcal L}$ must contain a pair of leaves from each pair of the three components of  $T-v$.  Here we provide a partial converse of this result by showing that if a set ${\mathcal L}$ of leaf pairs forms a cover  of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in ${\mathcal L}$. Moreover,  there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning 'triplet covers', and is relevant to a problem arising in evolutionary genomics.

scholarly journals Noneffective Regularity of Equality Languages and Bounded Delay Morphisms

2010 ◽  
Vol Vol. 12 no. 4 ◽  
Author(s):  
Juhani Karhumaki ◽  
Aleksi Saarela
Keyword(s):  
Special Issue ◽  
Bounded Delay ◽  
International Audience

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience We give an instance of a class of morphisms for which it is easy to prove that their equality set is regular, but its emptiness is still undecidable. The class is that of bounded delay 2 morphisms.

scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

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